This sentence could also be translated as "I am going by train right there"
Theoretically yes, but it would require some particular intonation. "Jedu vlakem právě tam." is much more natural. Not sure if we should accept that.
You should as we don't hear any intonation. Moreover, it's the movement section of the course, so "right there" is more thematic.
Why not, "I am going there right now by train"?
I am going there by train now, neuznava. Now je nyni, prave, nebo 'prave' je pouze right now?
Yes, but právě means emphatically "right now" (apart from other uses).